Final answer:
The characteristic equation for the associated homogeneous differential equation y'' + 6y' + 9y = 0 is r^2 + 6r + 9 = 0. Finding the roots of this equation will guide us to the solution of the homogeneous part, while the initial conditions y(0)=0 and y'(0)=5 will help determine the particular solution.
Step-by-step explanation:
To address the student's problem of solving a nonhomogeneous differential equation using the method of undetermined coefficients, it's necessary to first write the characteristic equation of the associated homogeneous equation. For the given equation, y'' + 6y' + 9y = 6te⁻³ᵗ + 15 - 18t, the associated homogeneous equation would be y'' + 6y' + 9y = 0. The characteristic equation for this differential equation is derived by replacing y with e^{rt} (where r is the rate of exponential growth or decay) and setting the equation equal to zero, which results in r^2 + 6r + 9 = 0.
The next steps include factoring or using the quadratic formula to solve for the roots of the characteristic equation. These roots are crucial as they determine the form of the general solution to the homogenous equation. The student's problem also includes initial conditions: y(0)=0 and y'(0)=5, which are necessary to determine the specific solution that satisfies these conditions.